Moderation hypothesis (continuous moderator, categorical IV)

Question

for my research I have a hypothesis containing a categorical IV (positive or negative article that the participants have to read) and a continuous moderator (ideology level of the participant). The x axis is the evaluation that participants are giving (by answering different questions on a 7-point-scala). On this following graph, you can see the interaction I'm suspecting (I further elaborate the concept in the last paragraph).

My question is, how can I write this down as a hypothesis if the moderator is continuous? It's clear to me how to do it when it's categorical and somehow I just seem to find examples of categorical moderators. I'm aware that the question might be really simple, but I'm really having trouble visualizing all this, as I imagine that I can't speak of 'low', 'moderate' and 'high' levels in my hypothesis? (again, the graph is only for visualizing, the moderator is continuous, even if I have 'divided' it in the graph). How do I highlight this interaction that is happening (or that I'm suspecting that it will happen) at the mid-level of ideology? If I'm just saying

I hope my question is clear and not too stupid :-)

Just a bit of context to understand my research: Participants will either read a positive or a negative description of the situation in a country before colonialism. For those having a middle level of ideology (so they are neither in favor of it, nor totally against it), they will evaluate colonialism more positive when they have read a negative article about the time before colonialism than when they have read a positive article. Now I suspect ideology to moderate this effect, when it's either on a low or a high level. For example, I suspect that participants with a high level of ideology will contest the article if it talks positively about the past (the past BEFORE the colonisation began). As a result, they will evaluate the colonial past HIGHER than in the situation where they read a negative article (and vice versa for the ones with a low ideology). How can I voice this without 'categorizing' my continuous variable with terms as low, middle, high?

Answer 1

What you're expecting is an interaction between article (pos or neg) and the quadratic effect of ideology. In other words, you're expecting that the curve of the ideology effect for positive article participants will differ from the curve for negative article participants.

Here's a model that would capture that hypothesis:

$DV = b0 + b1*ideology + b2*ideology^2 + b3*article + b4*article*ideology + b5*article*ideology^2 + e$

I'm assuming you'll dummy code article, and that you'll treat ideology as continuous. Your main hypothesis is that $b5$ will be significantly different from zero. From your plot, it looks like you're also expecting a linear effect of ideology (significant b1), and that you don't expect the linear effect of ideology to differ by article (non-sig b4), and you don't expect an overall main effect of article (non-sig b3). But those other aspects of the graph may not be central to your theory. You can run this model as multiple regression in your favorite statistical software.

Note that since you have seven levels of ideology, you would in theory be able to test polynomial terms up to a power of 6. In practice, I find anything about a cubic effect is pretty difficult to interpret usefully (especially when it's involved in interactions). You may want to test for those higher levels, though, to see if there's any non-linear effect you're not adequately capturing with the linear and quadratic effects. And, as always, plot and examine your residuals.